Set Notation and the Real Numbers Oh, and some stuff on functions, too 1
Elementary Set Theory Vocabulary: Set Element Subset Union Intersection Set Difference Disjoint A intersects B Empty set or null set Symbols: a A {x x is........ } A B or A B A B A B Ø A B or A B AxB 2
Functions Vocabulary: Domain Range Image Identity function Inverse function Pre Image or Inverse Image Composition of Functions One to one Onto One to one correspondence Symbols: f(a) (a is an element) f(a) (A is a set) f 1 1 (Inverse function) f 1 (B) (B is a set) f 1 (b) (b is an element) f g 3
Functions For us, a function from a set A to a set B is a rule that assigns to each element x of A a unique element f(x) of B. The set A is the domain of the function, the set B is the range. For us, the domain and range are part of the definition of a function. Thus we would not say that f(x)=3x+7 defines a function until the domain and range were specified. 4
Functions 2 Thus, the function f : defined by f () x = x is not the same as the function g: [0, ) defined by gx 2 () = x. This is the real point of a function being onto or not; the function f is not onto, while the function g is onto. (So for us, the answer to the typical calculus problem that asks, what is the range? is, You tell me! ) 5
We use the real numbers in the axioms of our particular brand of geometry. For the most part, our ordinary everyday knowledge of real numbers (e.g. knowing how to order them and do arithmetic with them) will do. However, there are a few places where we will rely on their deeper properties. We will list a few of them here. 6
Trichotomy: Given any two real numbers x and y, exactly one of the following is true: x < y x = y, or x > y. This is equivalent to saying that every real number is either positive, negative or 0. We will use this when we prove that linear pairs are supplementary. 7
Density: Between every two real numbers a and b, there is a rational number between a and b, and there is an irrational number between a and b. This is equivalent to saying that both the rational numbers and irrational numbers are dense in the real numbers. 8
Archimedean Property: If M and ε are any two positive real numbers, then there exists a positive integer n such that nε > M. If you think of M as very big and ε as very small, this says simply that if you take enough (i.e. n) copies of something small (ε), you can get as big as you want (M). We will use this later in proving facts about similarity. il i 9
Completeness: Thecompleteness propertyofof the real numbers has several different equivalent formulations, some of which are better suited for some purposes than others. We need a couple of definitions: Definition: A number b is called an upper bound for a set A of real numbers provided x a for every real number a in A. 10
Definition: A number l is called the least upper bound for a set A provided it is an upper bound for A and l b for every other upper bound b of A. Now we can state a couple of equivalent versions of completeness: 11
(Least Upper Bound Property): Every nonempty set of real numbers that has an upper bound has a least upper bound. Every bounded increasing sequence of real numbers converges. Every Cauchy sequence of real numbers converges. 12
This basic property of real numbers assures us that limits exist, and is the foundation for calculus. We ll use these properties a few times, e.g. in the proof of circular continuity and theproof of the Saccheri Legendre Theorem. 13